Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 6930*x^7 - 1152111*x^6 - 702950*x^5 + 20667206*x^4 + 25065130*x^3 - 101392810*x^2 - 59815601*x + 123164081)
gp: K = bnfinit(y^12 - y^11 - 285*y^10 + 285*y^9 + 27236*y^8 - 6930*y^7 - 1152111*y^6 - 702950*y^5 + 20667206*y^4 + 25065130*y^3 - 101392810*y^2 - 59815601*y + 123164081, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 6930*x^7 - 1152111*x^6 - 702950*x^5 + 20667206*x^4 + 25065130*x^3 - 101392810*x^2 - 59815601*x + 123164081);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 6930*x^7 - 1152111*x^6 - 702950*x^5 + 20667206*x^4 + 25065130*x^3 - 101392810*x^2 - 59815601*x + 123164081)
\( x^{12} - x^{11} - 285 x^{10} + 285 x^{9} + 27236 x^{8} - 6930 x^{7} - 1152111 x^{6} - 702950 x^{5} + \cdots + 123164081 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(84489053066670461041015625\)
\(\medspace = 5^{9}\cdot 13^{11}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(144.73\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $5^{3/4}13^{11/12}17^{1/2}\approx 144.7327131886099$ | ||
| Ramified primes: |
\(5\), \(13\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{65}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1105=5\cdot 13\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1105}(256,·)$, $\chi_{1105}(1,·)$, $\chi_{1105}(67,·)$, $\chi_{1105}(324,·)$, $\chi_{1105}(69,·)$, $\chi_{1105}(1089,·)$, $\chi_{1105}(33,·)$, $\chi_{1105}(713,·)$, $\chi_{1105}(747,·)$, $\chi_{1105}(203,·)$, $\chi_{1105}(341,·)$, $\chi_{1105}(577,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}$, $\frac{1}{20}a^{10}-\frac{1}{10}a^{9}+\frac{3}{20}a^{8}-\frac{1}{4}a^{7}+\frac{1}{5}a^{6}+\frac{3}{10}a^{5}-\frac{2}{5}a^{4}+\frac{9}{20}a^{3}+\frac{1}{5}a^{2}+\frac{1}{4}a+\frac{9}{20}$, $\frac{1}{26\!\cdots\!20}a^{11}+\frac{21\!\cdots\!44}{13\!\cdots\!71}a^{10}-\frac{55\!\cdots\!94}{65\!\cdots\!55}a^{9}+\frac{46\!\cdots\!14}{65\!\cdots\!55}a^{8}+\frac{86\!\cdots\!81}{65\!\cdots\!55}a^{7}+\frac{65\!\cdots\!29}{26\!\cdots\!20}a^{6}+\frac{15\!\cdots\!09}{26\!\cdots\!20}a^{5}-\frac{37\!\cdots\!21}{13\!\cdots\!10}a^{4}+\frac{75\!\cdots\!97}{26\!\cdots\!20}a^{3}+\frac{14\!\cdots\!63}{26\!\cdots\!20}a^{2}-\frac{11\!\cdots\!01}{26\!\cdots\!20}a-\frac{30\!\cdots\!07}{26\!\cdots\!20}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{10}$, which has order $20$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{37\!\cdots\!95}{10\!\cdots\!71}a^{11}-\frac{21\!\cdots\!46}{10\!\cdots\!71}a^{10}-\frac{38\!\cdots\!45}{40\!\cdots\!84}a^{9}+\frac{22\!\cdots\!35}{40\!\cdots\!84}a^{8}+\frac{73\!\cdots\!50}{10\!\cdots\!71}a^{7}-\frac{15\!\cdots\!15}{40\!\cdots\!84}a^{6}-\frac{96\!\cdots\!13}{40\!\cdots\!84}a^{5}+\frac{37\!\cdots\!75}{40\!\cdots\!84}a^{4}+\frac{12\!\cdots\!65}{40\!\cdots\!84}a^{3}-\frac{13\!\cdots\!15}{20\!\cdots\!42}a^{2}-\frac{12\!\cdots\!95}{20\!\cdots\!42}a+\frac{38\!\cdots\!43}{40\!\cdots\!84}$, $\frac{39\!\cdots\!95}{40\!\cdots\!84}a^{11}-\frac{23\!\cdots\!65}{40\!\cdots\!84}a^{10}-\frac{10\!\cdots\!45}{40\!\cdots\!84}a^{9}+\frac{15\!\cdots\!70}{10\!\cdots\!71}a^{8}+\frac{79\!\cdots\!55}{40\!\cdots\!84}a^{7}-\frac{10\!\cdots\!45}{10\!\cdots\!71}a^{6}-\frac{65\!\cdots\!84}{10\!\cdots\!71}a^{5}+\frac{98\!\cdots\!65}{40\!\cdots\!84}a^{4}+\frac{35\!\cdots\!75}{40\!\cdots\!84}a^{3}-\frac{69\!\cdots\!15}{40\!\cdots\!84}a^{2}-\frac{35\!\cdots\!95}{20\!\cdots\!42}a+\frac{10\!\cdots\!75}{40\!\cdots\!84}$, $\frac{86\!\cdots\!39}{26\!\cdots\!20}a^{11}-\frac{50\!\cdots\!21}{26\!\cdots\!20}a^{10}-\frac{11\!\cdots\!61}{13\!\cdots\!10}a^{9}+\frac{13\!\cdots\!21}{26\!\cdots\!20}a^{8}+\frac{17\!\cdots\!31}{26\!\cdots\!20}a^{7}-\frac{89\!\cdots\!03}{26\!\cdots\!20}a^{6}-\frac{11\!\cdots\!89}{52\!\cdots\!84}a^{5}+\frac{10\!\cdots\!26}{13\!\cdots\!71}a^{4}+\frac{18\!\cdots\!01}{65\!\cdots\!55}a^{3}-\frac{15\!\cdots\!17}{26\!\cdots\!20}a^{2}-\frac{73\!\cdots\!47}{13\!\cdots\!10}a+\frac{11\!\cdots\!99}{13\!\cdots\!10}$, $\frac{27\!\cdots\!17}{26\!\cdots\!20}a^{11}-\frac{16\!\cdots\!03}{26\!\cdots\!20}a^{10}-\frac{70\!\cdots\!31}{26\!\cdots\!20}a^{9}+\frac{20\!\cdots\!69}{13\!\cdots\!10}a^{8}+\frac{53\!\cdots\!43}{26\!\cdots\!20}a^{7}-\frac{70\!\cdots\!91}{65\!\cdots\!55}a^{6}-\frac{88\!\cdots\!41}{13\!\cdots\!71}a^{5}+\frac{13\!\cdots\!51}{52\!\cdots\!84}a^{4}+\frac{23\!\cdots\!97}{26\!\cdots\!20}a^{3}-\frac{47\!\cdots\!21}{26\!\cdots\!20}a^{2}-\frac{11\!\cdots\!23}{65\!\cdots\!55}a+\frac{69\!\cdots\!79}{26\!\cdots\!20}$, $\frac{26\!\cdots\!39}{26\!\cdots\!20}a^{11}-\frac{75\!\cdots\!03}{13\!\cdots\!10}a^{10}-\frac{69\!\cdots\!57}{26\!\cdots\!20}a^{9}+\frac{39\!\cdots\!41}{26\!\cdots\!20}a^{8}+\frac{13\!\cdots\!79}{65\!\cdots\!55}a^{7}-\frac{68\!\cdots\!62}{65\!\cdots\!55}a^{6}-\frac{18\!\cdots\!09}{26\!\cdots\!42}a^{5}+\frac{13\!\cdots\!49}{52\!\cdots\!84}a^{4}+\frac{12\!\cdots\!67}{13\!\cdots\!10}a^{3}-\frac{47\!\cdots\!77}{26\!\cdots\!20}a^{2}-\frac{50\!\cdots\!19}{26\!\cdots\!20}a+\frac{35\!\cdots\!39}{13\!\cdots\!10}$, $\frac{54\!\cdots\!77}{26\!\cdots\!20}a^{11}-\frac{81\!\cdots\!07}{65\!\cdots\!55}a^{10}-\frac{35\!\cdots\!09}{65\!\cdots\!55}a^{9}+\frac{42\!\cdots\!29}{13\!\cdots\!10}a^{8}+\frac{27\!\cdots\!87}{65\!\cdots\!55}a^{7}-\frac{57\!\cdots\!19}{26\!\cdots\!20}a^{6}-\frac{71\!\cdots\!07}{52\!\cdots\!84}a^{5}+\frac{13\!\cdots\!39}{26\!\cdots\!42}a^{4}+\frac{48\!\cdots\!17}{26\!\cdots\!20}a^{3}-\frac{99\!\cdots\!41}{26\!\cdots\!20}a^{2}-\frac{98\!\cdots\!47}{26\!\cdots\!20}a+\frac{14\!\cdots\!69}{26\!\cdots\!20}$, $\frac{20\!\cdots\!09}{26\!\cdots\!20}a^{11}-\frac{29\!\cdots\!43}{65\!\cdots\!55}a^{10}-\frac{26\!\cdots\!63}{13\!\cdots\!71}a^{9}+\frac{77\!\cdots\!97}{65\!\cdots\!55}a^{8}+\frac{10\!\cdots\!94}{65\!\cdots\!55}a^{7}-\frac{20\!\cdots\!67}{26\!\cdots\!20}a^{6}-\frac{13\!\cdots\!91}{26\!\cdots\!20}a^{5}+\frac{25\!\cdots\!79}{13\!\cdots\!10}a^{4}+\frac{35\!\cdots\!81}{52\!\cdots\!84}a^{3}-\frac{35\!\cdots\!21}{26\!\cdots\!20}a^{2}-\frac{33\!\cdots\!09}{26\!\cdots\!20}a+\frac{50\!\cdots\!09}{26\!\cdots\!20}$, $\frac{16\!\cdots\!51}{26\!\cdots\!42}a^{11}-\frac{18\!\cdots\!08}{65\!\cdots\!55}a^{10}-\frac{43\!\cdots\!11}{26\!\cdots\!20}a^{9}+\frac{19\!\cdots\!79}{26\!\cdots\!20}a^{8}+\frac{18\!\cdots\!25}{13\!\cdots\!71}a^{7}-\frac{11\!\cdots\!13}{26\!\cdots\!20}a^{6}-\frac{13\!\cdots\!67}{26\!\cdots\!20}a^{5}+\frac{19\!\cdots\!91}{26\!\cdots\!20}a^{4}+\frac{20\!\cdots\!67}{26\!\cdots\!20}a^{3}+\frac{28\!\cdots\!51}{13\!\cdots\!10}a^{2}-\frac{23\!\cdots\!23}{13\!\cdots\!71}a-\frac{31\!\cdots\!03}{26\!\cdots\!20}$, $\frac{20\!\cdots\!99}{13\!\cdots\!10}a^{11}-\frac{32\!\cdots\!21}{65\!\cdots\!55}a^{10}-\frac{10\!\cdots\!85}{26\!\cdots\!42}a^{9}+\frac{16\!\cdots\!83}{13\!\cdots\!10}a^{8}+\frac{22\!\cdots\!18}{65\!\cdots\!55}a^{7}-\frac{91\!\cdots\!47}{13\!\cdots\!10}a^{6}-\frac{16\!\cdots\!61}{13\!\cdots\!10}a^{5}+\frac{41\!\cdots\!29}{65\!\cdots\!55}a^{4}+\frac{47\!\cdots\!81}{26\!\cdots\!42}a^{3}+\frac{11\!\cdots\!37}{65\!\cdots\!55}a^{2}-\frac{13\!\cdots\!07}{65\!\cdots\!55}a-\frac{26\!\cdots\!01}{13\!\cdots\!10}$, $\frac{17\!\cdots\!05}{52\!\cdots\!84}a^{11}-\frac{25\!\cdots\!81}{13\!\cdots\!10}a^{10}-\frac{11\!\cdots\!23}{13\!\cdots\!10}a^{9}+\frac{67\!\cdots\!47}{13\!\cdots\!10}a^{8}+\frac{17\!\cdots\!35}{26\!\cdots\!42}a^{7}-\frac{91\!\cdots\!13}{26\!\cdots\!20}a^{6}-\frac{58\!\cdots\!57}{26\!\cdots\!20}a^{5}+\frac{54\!\cdots\!59}{65\!\cdots\!55}a^{4}+\frac{78\!\cdots\!17}{26\!\cdots\!20}a^{3}-\frac{15\!\cdots\!33}{26\!\cdots\!20}a^{2}-\frac{31\!\cdots\!61}{52\!\cdots\!84}a+\frac{22\!\cdots\!97}{26\!\cdots\!20}$, $\frac{10\!\cdots\!21}{26\!\cdots\!20}a^{11}-\frac{62\!\cdots\!79}{26\!\cdots\!20}a^{10}-\frac{68\!\cdots\!12}{65\!\cdots\!55}a^{9}+\frac{16\!\cdots\!19}{26\!\cdots\!20}a^{8}+\frac{21\!\cdots\!49}{26\!\cdots\!20}a^{7}-\frac{10\!\cdots\!47}{26\!\cdots\!20}a^{6}-\frac{13\!\cdots\!05}{52\!\cdots\!84}a^{5}+\frac{26\!\cdots\!57}{26\!\cdots\!42}a^{4}+\frac{46\!\cdots\!43}{13\!\cdots\!10}a^{3}-\frac{18\!\cdots\!23}{26\!\cdots\!20}a^{2}-\frac{46\!\cdots\!89}{65\!\cdots\!55}a+\frac{67\!\cdots\!03}{65\!\cdots\!55}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 101052811.273 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 101052811.273 \cdot 20}{2\cdot\sqrt{84489053066670461041015625}}\cr\approx \mathstrut & 0.450306407485 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 6930*x^7 - 1152111*x^6 - 702950*x^5 + 20667206*x^4 + 25065130*x^3 - 101392810*x^2 - 59815601*x + 123164081)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 6930*x^7 - 1152111*x^6 - 702950*x^5 + 20667206*x^4 + 25065130*x^3 - 101392810*x^2 - 59815601*x + 123164081, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 6930*x^7 - 1152111*x^6 - 702950*x^5 + 20667206*x^4 + 25065130*x^3 - 101392810*x^2 - 59815601*x + 123164081);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 285*x^10 + 285*x^9 + 27236*x^8 - 6930*x^7 - 1152111*x^6 - 702950*x^5 + 20667206*x^4 + 25065130*x^3 - 101392810*x^2 - 59815601*x + 123164081);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{65}) \), 3.3.169.1, 4.4.79366625.2, 6.6.46411625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | R | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(13\)
| 13.12.11.11 | $x^{12} + 65$ | $12$ | $1$ | $11$ | $C_{12}$ | $[\ ]_{12}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |